Question

For what values of `x` is the function `f(x) = sin x + cos x` continuous?

Answer 1

It is continuous on the whole of `RR`

Explanation:

Given `f(x) = sin x + cos x`

If you are happy that `sin x` and `cos x` are both continuous functions, then this is the sum of two continuous functions and is therefore continuous.

On the other hand, if you want to prove it from basic principles, we can proceed as follows:

We find:

`f(x) = sin x + cos x`

`color(white)(f(x)) = sqrt(2)(sqrt(2)/2 sin x + sqrt(2)/2 cos x)`

`color(white)(f(x)) = sqrt(2)(sin x cos (pi/4) + cos x sin (pi/4))`

`color(white)(f(x)) = sqrt(2)sin (x+pi/4)`

Then:

`f(x+delta) = sqrt(2)sin (x + pi/4 + delta)`

`color(white)(f(x+delta)) = sqrt(2)(sin (x + pi/4) cos delta + cos (x + pi/4) sin delta)`

Note that:

`{ (lim_(delta->0) sin delta = 0), (lim_(delta->0) cos delta = 1) :}`

So:

`lim_(delta->0) f(x + delta) = lim_(delta->0) sqrt(2)(sin (x + pi/4) cos delta + cos (x + pi/4) sin delta)`

`color(white)(lim_(delta->0) f(x + delta)) = sqrt(2)(sin (x + pi/4) (1) + cos (x + pi/4) (0))`

`color(white)(lim_(delta->0) f(x + delta)) = sqrt(2)sin (x + pi/4)`

`color(white)(lim_(delta->0) f(x + delta)) = f(x)`

So `f(x)` is continuous at `x` for any `x in RR`